M '=- 



292 M, P. S. Girard on Navigable Canals, 



This value of u' substituted in the second equation 



gives 



S / B \ 



Substituting, in the same manner, the equivalent of u' and 

 u" in the equation 

 S 



• w'"=^-g:j:g-(D-(a: + M' + M")). 



it becomes 



_S_/B^-x)\ 

 B+S \(B4-S)2 )' 

 We shall find successively r 



S / B3 ^ 



""=B+^i(B+S?(^-^)j 



«' =B+"SV(B+Sy*~(^-^)J\ 

 Therefore the sum of the successive augmentations oh 

 the upper level B, that is to say 



S /, B .« 



(gfs)+(Bfs)+(Bfs)+--(Bfs)""')' 



Or, in other words, the whole elevation of the surface, 

 occasioned by the passage of a number n of boats, is ex- 

 pressed by a decreasing geometrical progression, the num- 

 ber of whose terms is ji, and whose ratio is 

 B 



B+S. 

 Whence we conclude that whenever the superficies of the 

 basin S is not so small, in proportion to the superficies B of 

 the upper level, as to be neglected, the rise of water occa-^ 

 sioned by the descent of a boat S, will always be propor- 

 tionate to a certain power of the fraction 



B 

 B+S ' 

 power whose value will always be so much the less as the 

 boat approaches the numerical close of the series of descend 

 ing boats. 



